A relation $mathcal{M}_{mathrm{SHS}tomathrm{LJ}}$ between the set of non-isomorphic sticky hard sphere clusters $mathcal{M}_mathrm{SHS}$ and the sets of local energy minima $mathcal{M}_{LJ}$ of the $(m,n)$-Lennard-Jones potential $V^mathrm{LJ}_{mn}(r) = frac{varepsilon}{n-m} [ m r^{-n} - n r^{-m} ]$ is established. The number of nonisomorphic stable clusters depends strongly and nontrivially on both $m$ and $n$, and increases exponentially with increasing cluster size $N$ for $N gtrsim 10$. While the map from $mathcal{M}_mathrm{SHS}to mathcal{M}_{mathrm{SHS}tomathrm{LJ}}$ is non-injective and non-surjective, the number of Lennard-Jones structures missing from the map is relatively small for cluster sizes up to $N=13$, and most of the missing structures correspond to energetically unfavourable minima even for fairly low $(m,n)$. Furthermore, even the softest Lennard-Jones potential predicts that the coordination of 13 spheres around a central sphere is problematic (the Gregory-Newton problem). A more realistic extended Lennard-Jones potential chosen from coupled-cluster calculations for a rare gas dimer leads to a substantial increase in the number of nonisomorphic clusters, even though the potential curve is very similar to a (6,12)-Lennard-Jones potential.
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